246 



Mr. G. W. Walker on the 



The variation in k'^ will thus be negative or positive 

 according as a^—1 > or < 1 — hJ^. The formula would be 

 invalid if {a^ — l)={l — bj'^). A brief examination shows 

 that with a finite breadth of needle a^—1 is >l — h^"^. 



Consider the function 



Then 

 and 



B/ 



. 1, l+/> h 



/=0 when /> = 0, 

 /= + (X) when Z> = 1, 

 'df_ a^-l + h'^-l 



^6- (l_^2j(^2_l)- 



Now iT^, will vanish when a^— 1 

 00 



l-h^ or ^^ = 2 



B/ 



Hence, if a^ is < 2, there is a real value for which ^ 



vanishes ; but if a^ is > 2, there is no real value. There are 

 thus two cases, A and B, as shown in the figure. 



Xow the root hJ is oiven bv the intersection of the curve 



(Jo 1/ 



^ 1, l + h h 



with the straight line 



f= 



IT § 



2 8 



